The turboprop, and especially the high-blade-count propfan, differs from the high-bypass turbofan principally in throwing away the draggy outer ducting. This makes it easier to obtain a greater diameter, moving more air slower, and hence higher efficiency. Fewer blades further increase aerodynamic efficiency. Tip speeds remain much the same, limited by the speed of sound. So why are turboprops significantly slower than high-bypass turbofans?
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Tip speeds do not actually stay the same, turbofan (and turbojet) blade tip speeds can exceed the speed of sound, up to Mach 1.3 (actually stated M1.4 in the link, my experience is M1.3 but also my professor used to mention M1.4) as seen in link. As they are enclosed, the negative effects of the shock are much smaller than if it were an open propeller. Open propellers (and also propfans) are also affected by aeroelastic effects in transonic speeds which are very hard to isolate with their high span-to-chord ratios (I don't like calling it aspect ratio for propellers). Not to mention the NOISE that an open propeller generates at transonic speeds.
Additionally, due to the fan being followed by a stator (which uses the rotational component of velocity added to the airflow by the rotor) and the converging fan duct geometry, turbofans accelerate the flow further than propellers do. The exhaust from the turbine is also faster than the fan and constitutes a greater fraction of the air ingested by the engine, which increases the average exhaust velocity of the engine.
Imagine a turboprop flying at Mach 0.8 and the propeller tip speed being limited to Mach 0.9 . This requires the propeller tip to have a tangential velocity of M0.41 (Pythagorean sum). On the other hand, at the same flight speed, a fan blade tip limited to M1.3 can still have a supersonic tangential velocity of M1.024 . It is obvious that a fan blade turning more than twice as fast can exert greater forces on the airflow.
The thrust is generated by accelerating airflow. We can either accelerate a lot of air by a little bit (think of a helicopter rotor) or a little air by a lot (afterburning turbojet). A relatively slow-moving propeller blade (due to the reasons above) can do little to accelerate airflow going through it, relative to a fan that turns much faster in an enclosure.
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$\begingroup$ Ah, so fan blades can go supersonic. That led me to an earlier question, which mine to a large extent duplicates: aviation.stackexchange.com/questions/22316/… Note that exit velocities and efficiency are not related to my question here. $\endgroup$ Commented Nov 21, 2020 at 13:55
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$\begingroup$ exit velocities are relevant, I was trying to elaborate how turbofans can accelerate airflow at M0.8 while a propeller can't. $\endgroup$ Commented Nov 21, 2020 at 13:57
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1$\begingroup$ If you want to know what happens if you try to make a turboprop do jet speeds, Tupolev tried that; suffice it to say, from all I have read, the results were un-bear-ably loud $\endgroup$ Commented Nov 22, 2020 at 18:45
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$\begingroup$ I see what you did there. The NK-12 is a beast. $\endgroup$ Commented Nov 22, 2020 at 18:47